Singularities of harmonic maps

The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order elliptic equation for its tension. A Hasimoto transformation is used to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy. A frame frame parallel-transported along the elastica is in general only quasi-periodic, resulting in time-dependent boundary conditions for the coordinates. This complication is addressed by a gauge transformation, after which a contraction mapping argument can be applied. Local existence and uniqueness of elastica solutions are established for initial data in suitable Sobolev spaces.

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“…(2.20) below). An alternate choice for the potential energy related to the Dirichlet form energies for harmonic maps [9,10,11,12,13]…”

Section: Related Geometrical and Physical Problemsmentioning

confidence: 99%

On the Cauchy Problem for a Dynamical Euler's Elastica

Burchard

Thomas

2003

Communications in Partial Differential Equations

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“…Currently our direct mapping method is not provably diffeomorphic, but our experimental results demonstrate that it can obtain smooth maps of good metric preserving quality. Theoretically it is also possible for the heat flow of harmonic maps to have singularities under boundary constraints (Hardt, 1997). Finite-time blow-ups of a heat flow that maps from a disk to the sphere were reported in (Chang et al, 1992).…”

Section: Quantitative Comparisons With a Parametric Mapping Methodsmentioning

confidence: 99%

Direct cortical mapping via solving partial differential equations on implicit surfaces

Shi

Thompson

Dinov

et al. 2007

Medical Image Analysis

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In this paper, we propose a novel approach for cortical mapping that computes a direct map between two cortical surfaces while satisfying constraints on sulcal landmark curves. By computing the map directly, we can avoid conventional intermediate parameterizations and help simplify the cortical mapping process. The direct map in our method is formulated as the minimizer of a flexible variational energy under landmark constraints. The energy can include both a harmonic term to ensure smoothness of the map and general data terms for the matching of geometric features. Starting from a properly designed initial map, we compute the map iteratively by solving a partial differential equation (PDE) defined on the source cortical surface. For numerical implementation, a set of adaptive numerical schemes are developed to extend the technique of solving PDEs on implicit surfaces such that landmark constraints are enforced. In our experiments, we show the flexibility of the direct mapping approach by computing smooth maps following landmark constraints from two different energies. We also quantitatively compare the metric preserving property of the direct mapping method with a parametric mapping method on a group of 30 subjects. Finally, we demonstrate the direct mapping method in the brain mapping applications of atlas construction and variability analysis.

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“…In higher dimensions the picture changes drastically. For d = 3 harmonic maps into the sphere are partially regular if they are energy minimizing or, more generally, stationary with respect to spatial variations [21,10,24]. Those results are sharp in the sense that the function x → x/|x| is an energy minimizing harmonic map into the sphere if d ≥ 3 [30,11,27,39], and there exist harmonic maps into S n−1 which are everywhere discontinuous if d, n ≥ 3 [36].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces

Bartels¹

2009

Math. Comp.

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Abstract. This article studies the numerical approximation of harmonic maps into surfaces, i.e., critical points for the Dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold. An iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised, and its global convergence to a discrete harmonic map is proved under general conditions. Weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations. Numerical simulations illustrate the performance of the algorithm for curved domains.

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Singularities of harmonic maps

Abstract: Abstract. This article surveys research on the existence, structure, behavior, and asymptotics of singularities of harmonic maps.

Cited by 76 publications

References 103 publications

On the Cauchy Problem for a Dynamical Euler's Elastica

On the Cauchy Problem for a Dynamical Euler's Elastica

Direct cortical mapping via solving partial differential equations on implicit surfaces

Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces

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